# Quizz vector spaces --- Introduction ---

This module contains 11 training exercises on basic notions of vector spaces.

Students may use them as a training tool for a better memorization of the lesson, while teachers may put them into a work sheet with activated time limit, so that proficiency can be tested.

### Two subsets

Let be a vector space. We have two subsets of , and , having respectively and elements. Answer:
• If , then .
• If , then .

Fill in:

Fill in:

### Dependence

Let be a vector space, a non-empty subset.

• . True?
• . True?

### Dimension and elements

Let be a vector space, a finite subset that . If , .

### Dim subspace by system

Let E be a sub-vector space of R defined by a homogeneous linear system. This system is composed of equations, and the rank of the coefficient matrix of this system is equal to . What is the dimension of E?

### Generated subspace

Let be a vector subspace of generated by a subset of elements. What can be said of the dimension of ?

dim( ) is equal to .

### Inclusion

Let be a vector space, , two distinct finite subsets with .

• . True?
• . True?

### Set

Let be a vector space of dimension , a subset of elements.

• Can be linearly independent?
• Can generate ?
• Can be a basis of ?

### Set and basis

Let be a vector space of dimension , a subset of elements. Among the following properties, which ones  ?
 . . . .

### Generating subsets

Let be a vector space generated by a set = { }. Given that there is a relation
,
which of the following subsets already generate ?
• }
• }
• }
• }
• }

Other exercises on: vector spaces   linear algebra

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• Description: elementary questions on vector spaces. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games
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